An action potential is often introduced as a sequence of named stages: depolarization, overshoot, repolarization, and afterhyperpolarization. Those labels are useful descriptions of the voltage trace, but they do not explain why the trace has that shape. The Hodgkin–Huxley model supplies the mechanism. Membrane voltage changes the sodium and potassium conductances; those conductances change the ionic currents; and the currents change membrane voltage again. A spike is therefore not a command passed through a row of stages. It is a trajectory generated by interacting positive and negative feedback processes.
This interactive laboratory follows that logic from its simplest starting point. It begins with a passive membrane that can only charge and discharge. It then uses voltage clamp to reveal the hidden gating processes, releases the clamp to generate an action potential, shows how the gates create refractoriness, and finally places resting and repetitive firing within the language of dynamical systems.
What this laboratory is designed to teach
The simulation is organized around one central question:
At this instant, which currents are flowing, what are the gates doing, and why is membrane voltage moving in this direction?
By the end of the sequence, you should be able to explain why a passive response is graded, why voltage clamp is needed to measure voltage-dependent currents, how fast sodium activation creates regenerative excitation, how slower sodium inactivation and potassium activation terminate the spike, why excitability depends on recent history, and why a single spike is not the same dynamical object as repetitive firing.
The module also makes several distinctions that are easily blurred in a static diagram:
- current is not voltage;
- conductance is not current;
- channel activation is not the same as ionic driving force;
- the voltage at one moment does not fully specify the membrane state;
- a large transient excursion is not a limit cycle;
- and an action-potential phase is a period of changing dominance among overlapping currents, not a discrete processing stage.
The model in one view
The active-membrane modes use the classical space-clamped Hodgkin–Huxley model. The membrane state is
\[
\mathbf{X}(t)=[V(t),m(t),h(t),n(t)],
\]
where \(V\) is membrane voltage, \(m\) is sodium activation, \(h\) is sodium availability or inactivation, and \(n\) is potassium activation. The voltage equation is
\[
C_m\frac{dV}{dt}
=
I_{\mathrm{app}}
-I_{\mathrm{Na}}
-I_{\mathrm{K}}
-I_{\mathrm{L}}.
\]
The ionic currents are
\[
I_{\mathrm{Na}}
=
\bar g_{\mathrm{Na}}m^3h(V-E_{\mathrm{Na}}),
\]
\[
I_{\mathrm{K}}
=
\bar g_{\mathrm{K}}n^4(V-E_{\mathrm{K}}),
\]
and
\[
I_{\mathrm{L}}
=
g_{\mathrm{L}}(V-E_{\mathrm{L}}).
\]
The maximum conductances and reversal potentials are fixed parameters. The factors \(m^3h\) and \(n^4\) determine how much of the sodium and potassium conductance is currently available. Each gating variable changes according to the present voltage:
\[
\frac{dx}{dt}
=
\alpha_x(V)(1-x)-\beta_x(V)x,
\]
or, equivalently,
\[
\frac{dx}{dt}
=
\frac{x_\infty(V)-x}{\tau_x(V)}.
\]
The second form is especially useful for reading the simulation. At each voltage, a gate has a voltage-dependent target \(x_\infty(V)\) and a voltage-dependent time constant \(\tau_x(V)\). The present gate value does not jump instantly to its target. It approaches that target at a characteristic rate.
The drawings should not be interpreted as one literal sodium channel and one literal potassium channel. The variables \(m\), \(h\), and \(n\) are deterministic descriptions of a population-average conductance. The channel apertures and moving ions visualize the model’s aggregate conductance and current; they do not simulate the random opening of individual channel molecules.
Parameters and sign convention
The active modes use the canonical squid giant-axon values
\[
C_m=1\ \mu\mathrm{F/cm^2},
\qquad
\bar g_{\mathrm{Na}}=120\ \mathrm{mS/cm^2},
\]
\[
\bar g_{\mathrm{K}}=36\ \mathrm{mS/cm^2},
\qquad
g_{\mathrm{L}}=0.3\ \mathrm{mS/cm^2},
\]
\[
E_{\mathrm{Na}}=+50\ \mathrm{mV},
\qquad
E_{\mathrm{K}}=-77\ \mathrm{mV},
\qquad
E_{\mathrm{L}}=-54.387\ \mathrm{mV}.
\]
Holding current
When a preset removes a conductance, \(-65\) mV generally stops being an equilibrium. With \(\bar g_{\mathrm{K}}\) set to zero, for instance, the unforced resting potential of the remaining system is about \(-0.6\) mV, because the resting potassium conductance \(\bar g_{\mathrm{K}}n_\infty^4 \approx 0.37\ \mathrm{mS/cm^2}\) is larger than the leak conductance it leaves behind.
The conductance-block and altered-gradient presets therefore inject a small constant holding current that keeps the membrane at \(-65\) mV before the stimulus, exactly as an experimenter would. It is reported in the measurement boxes and is visible as a constant offset in the applied-current trace. For the unmodified model it is essentially zero; blocking \(\bar g_{\mathrm{K}}\) requires \(-4.40\ \mu\mathrm{A/cm^2}\), blocking \(\bar g_{\mathrm{Na}}\) requires \(+1.22\), and reducing \(E_{\mathrm{Na}}\) requires \(+0.53\).
Without this compensation the membrane drifts away from \(-65\) mV as soon as the trial begins, and a “block potassium” experiment fires an action potential before the stimulus arrives. One preset deliberately shows that; see section 3.
The current convention is important. Ionic current is defined as
\[
I_{\mathrm{ion}}=g(V-E),
\]
so positive ionic current is outward. Under current clamp, the voltage equation subtracts the ionic currents. An inward sodium current is therefore negative as an ionic current but contributes positively to \(dV/dt\) and depolarizes the membrane. The current-balance panel handles this bookkeeping for you: in current-clamp modes it displays each current by its contribution to voltage change, whereas in voltage-clamp mode it displays the measured membrane currents using the outward-positive convention.
How to operate the module
The module opens paused. Select a mode and experiment, then press Play. Every display is synchronized to the same model time and the same numerical state.
Playback controls
Play/Pause starts and stops the visual playback. Restart returns to the beginning without changing the selected experiment. ◀ Step and Step ▶ move backward and forward by a small amount, which is what you want near spike initiation, the spike peak, or a second refractory-period pulse: you can cross the same moment repeatedly in both directions. The timeline can be dragged to any instant. Playback speed ranges from 0.25× to 3×; slowing the display is usually more useful than changing the numerical model.
Keyboard shortcuts
1–5 |
Switch mode |
Space |
Play / pause |
← / → |
Step back / forward (hold Shift for five steps) |
Home / End |
Jump to the start or end of the trial |
[ / ] |
Previous / next experiment in this mode |
C |
Pin the current run for comparison |
R |
Restart |
P |
Presentation mode |
Presentation mode hides the explanatory sidebar text, the measurement log, and the reference panel, and enlarges the animation and the plots. The mode description, controls, state readout, and measurements stay visible. It is intended for projection.
Comparing two runs
Every plot can display one pinned run behind the live one as a dashed grey trace, in both the time series and the state projection. The previous run is pinned automatically whenever you change experiment, so “run A, run B, compare” needs no extra step; Pin run for comparison and Clear comparison give manual control.
While a comparison is showing, each axis is scaled to the union of both runs. Without that, the axis would silently rescale between them and two traces of very different size could look identical.
Measurement log
Each run you open is recorded in the Measurement log beneath the plots, with its four measured values and whether it passed its numerical check. Copy as CSV puts the whole table on the clipboard. The guided sequence below asks you to record values across several presets; the log means you do not have to copy them by hand.
Experiment and trace selectors
Each mode contains several validated experiment presets. Changing the Experiment menu recomputes the trajectory from the beginning. The explanatory sentence beneath the menu states exactly what was changed.
The Trace view menu changes only the display, not the model. Depending on the mode, you can show voltage and stimulus, the gating variables, conductances, ionic currents, or a combined view. The moving vertical cursor marks the same instant shown in the membrane animation and state plot.
State readout, measurements, and validation
The state boxes report the current values of \(V\), \(\mathrm{d}V/\mathrm{d}t\), \(m\), \(h\), and \(n\). In the passive control condition the voltage-gated variables are absent and are shown as dashes. The rate of change is reported alongside voltage because several parts of this guide ask you to watch it directly; it is also available as a trace under Voltage and dV/dt.
The four measurement boxes change with the mode. They report quantities such as the membrane time constant, peak ionic currents, spike latency, the holding current, gate values at the second pulse, spike period, or period variability. These values describe the complete simulated trajectory, not merely the moment currently displayed.
A green validation badge checks that the selected preset produced its intended numerical behavior. It verifies that a passive response returns toward rest, that the early sodium current precedes the delayed potassium current under voltage clamp, that repetitive firing has a stable period, and — for the current-clamp modes — three things worth stating explicitly:
- that the membrane was at equilibrium before the stimulus, so the baseline did not drift;
- that any action potential occurred after the stimulus rather than before it;
- that the number of spikes matches the preset’s claim.
A spike is identified by the membrane’s own regenerative current, not by crossing a fixed voltage line. A detector keyed to 0 mV would be unable to evaluate the reduced-\(E_{\mathrm{Na}}\) preset at all, because with \(E_{\mathrm{Na}}=0\) mV the voltage can never reach 0.
How to read the synchronized displays
Membrane mechanism
The upper display is a functional membrane patch. The voltage ruler on the left shows the current membrane voltage. Markers on the right identify the sodium, potassium, and leak reversal potentials. These reversal potentials are not thresholds. They are the voltages at which the corresponding ionic current changes direction.
The membrane surfaces display separated charge. The current source or voltage-clamp assembly appears on the left. In current clamp, the applied current is controlled and voltage is free to move. In voltage clamp, voltage is commanded and the clamp current is measured.
The sodium aperture follows \(m^3\), while the separate sodium inactivation element follows \(h\). The potassium aperture follows \(n^4\). A wide aperture means high conductance, but it does not by itself imply a large current. Current also depends on the driving force \(V-E\). The moving ion markers indicate the direction and approximate magnitude of the modeled current.
The status chip at the upper right names the current regime—for example, membrane charging, regenerative sodium feedback, net current crossing zero, potassium-dominated repolarization, afterhyperpolarization, or recovery. These labels summarize the continuously changing balance of the model; they should not be read as independent boxes through which the membrane passes.
Gating state
Each gate bar contains two indicators:
- the filled portion is the present value of the gate;
- the dark tick is the target value \(x_\infty(V)\) at the current voltage.
A voltage change can move the target immediately, while the gate itself moves gradually. This is particularly clear under voltage clamp. The \(m\) target jumps upward during depolarization, and \(m\) rapidly approaches it. The \(h\) target moves downward and the \(n\) target moves upward, but those gates respond more slowly. The action potential depends on this separation of time scales.
Current balance
In current-clamp modes, the current panel asks Why is voltage changing now? The injected-current bar and each ionic-current bar are plotted according to whether they contribute to depolarization or hyperpolarization. Their sum is
\[
C_m\frac{dV}{dt}.
\]
When the net value is positive, voltage rises. When it is negative, voltage falls. At the peak of an ordinary action potential, the net current passes through zero: inward and outward contributions balance momentarily even though substantial ionic currents may still be flowing.
In voltage clamp, the panel asks What current does the clamp measure? Here the bars retain the outward-positive ionic-current convention. The clamp supplies the current needed to impose the voltage command while the ionic conductances evolve beneath it.
Electrical state through time
The lower-left graph displays the selected traces. The traces are not separate simulations. They are different views of the same trajectory. Event lines mark the onset of current pulses, voltage commands, or manual nudges. The moving dots and vertical cursor show the present instant.
Use the trace selector rather than trying to interpret every variable at once. Voltage and stimulus reveal whether a response is graded or regenerative. Gating traces reveal the temporal ordering of \(m\), \(h\), and \(n\). Conductance traces show the consequences of the gates. Current traces combine conductance with driving force.
Voltage dependence of the gates
Beneath the two plots, a panel shows \(x_\infty(V)\) and \(\tau_x(V)\) for all three gates across the full voltage range, with a vertical line at the present membrane voltage and a dot on each curve.
This is the panel that makes the gate bars legible. The tick on each gate bar is a single point read off these curves at the current voltage. Watching the line sweep left and right during a spike shows why \(m\) can reach its target almost immediately while \(h\) and \(n\) lag: at \(-65\) mV, \(\tau_m\) is a fraction of a millisecond while \(\tau_h\) and \(\tau_n\) are several milliseconds, and the ratio changes as voltage moves.
State trajectory
The lower-right graph asks how the joint state evolves rather than how one variable changes through time. The coordinates depend on the mode:
- Passive membrane: \(V\) versus \(dV/dt\).
- Voltage clamp: \(m\) versus \(n\), with color representing \(h\).
- Active current-clamp modes: \(V\) versus \(n\), with color representing \(h\) and point size representing \(m\).
The full active Hodgkin–Huxley model has four state variables. The \(V\)-versus-\(n\) graph is therefore a projection of a four-dimensional trajectory, not the complete phase space. Two plotted points can occupy the same \(V\) and \(n\) coordinates while differing in \(m\) or \(h\).
1. Passive membrane: graded charging and discharging
Select Passive membrane. This control condition removes the voltage-gated sodium and potassium conductances. The membrane obeys
\[
C_m\frac{dV}{dt}
=
I_{\mathrm{app}}(t)-g_{\mathrm{pass}}(V-E_{\mathrm{rest}}),
\]
with membrane time constant
\[
\tau_m=\frac{C_m}{g_{\mathrm{pass}}}.
\]
A square current pulse begins at 5 ms and ends at 20 ms. Current begins immediately, but voltage cannot jump because the membrane capacitance must charge. As voltage moves away from \(E_{\mathrm{rest}}\), the opposing leak current grows. After the pulse ends, leak discharges the membrane back toward rest.
Baseline membrane
Choose Baseline membrane and press Play. The model uses \(C_m=1\ \mu\mathrm{F/cm^2}\), \(g_{\mathrm{pass}}=0.10\ \mathrm{mS/cm^2}\), and a \(2.5\ \mu\mathrm{A/cm^2}\) pulse. The time constant is 10 ms. Watch the applied-current bar turn on before voltage reaches its new level. The passive phase-plane trajectory leaves the resting point, curves toward the pulse-driven equilibrium, and then returns after the pulse ends.
The response is graded. Its amplitude depends continuously on the input. There is no regenerative event and no all-or-none boundary.
Larger capacitance
Choose Larger capacitance. Capacitance is doubled while leak conductance and current are unchanged. Predict the result before pressing Play.
The steady voltage that would eventually be reached during a sufficiently long pulse is unchanged because the steady-state condition is determined by \(I/g\). The approach is slower because \(\tau_m=C_m/g\) has doubled. With the finite pulse used here, the membrane has less time to approach its eventual steady level before the current turns off.
This manipulation isolates the role of capacitance: it changes the rate at which voltage changes, not the steady voltage implied by the balance of constant current and leak.
Larger leak conductance
Choose Larger leak conductance. Doubling leak has two effects at once. It lowers the input resistance, so the same current produces a smaller steady voltage displacement, and it shortens the membrane time constant, so charging and discharging are faster.
This is an important contrast with increasing capacitance. Both manipulations concern membrane time, but they do not have the same effect on response amplitude.
Stronger current pulse
Choose Stronger current pulse. The response becomes larger, but it remains graded. Nothing in a passive membrane converts increasing input into a stereotyped action potential. The comparison establishes the mechanism that must be added: voltage-dependent positive feedback.
A membrane is not an instantaneous voltage meter. It is a capacitor in parallel with conductive pathways. Applied current changes stored charge, and voltage evolves according to the net current. Passive dynamics explain delay, smoothing, and return to rest, but not regenerative excitation.
2. Voltage clamp: revealing the hidden conductances
Select Voltage clamp. In this mode, voltage is no longer an outcome. It is the experimental command. The clamp imposes a holding voltage, steps to a test voltage from 2 to 10 ms, and then returns to the holding voltage. The gates continue to evolve according to the commanded voltage, and the clamp supplies whatever current is necessary to prevent those ionic currents from changing \(V\).
For a gating variable \(x\),
\[
\frac{dx}{dt}
=
\frac{x_\infty(V_{\mathrm{cmd}})-x}{\tau_x(V_{\mathrm{cmd}})}.
\]
The measured clamp current is
\[
I_{\mathrm{clamp}}
=
C_m\frac{dV_{\mathrm{cmd}}}{dt}
+I_{\mathrm{Na}}+I_{\mathrm{K}}+I_{\mathrm{L}}.
\]
The command changes over a short finite ramp so that the capacitive transient remains finite in the numerical display.
Large depolarizing step
Begin with Large depolarizing step (−65 → −20 mV). Select Gating variables and press Play. At the command step, the target for \(m\) jumps upward and \(m\) rapidly follows. The target for \(h\) falls, while the target for \(n\) rises. Their slower movement produces a changing mixture of conductances even though voltage is held constant.
Now select Ionic currents and restart. The early sodium current is inward and transient. It appears quickly because \(m\) activates quickly, then declines as \(h\) decreases. The potassium current develops more slowly because \(n\) activates more slowly and remains outward at the test voltage. The measurement boxes report the peak currents and their times; the sodium peak should precede the potassium peak.
This sequence shows why the action-potential currents could not be inferred from the voltage trace alone. Holding voltage fixed separates the time-dependent conductance changes from the voltage changes those conductances would otherwise produce.
Small depolarizing step
Choose Small depolarizing step (−65 → −40 mV). The same gating rules apply, but the target values and driving forces differ. Compare the size and time course of the currents with those produced by the larger step. Voltage dependence is continuous: the channels do not wait for one universal threshold voltage before responding.
Sodium and potassium conductance blocks
Choose Block sodium conductance. The voltage command is unchanged, but \(\bar g_{\mathrm{Na}}\) is set to zero. The early sodium component disappears while the delayed potassium current remains.
Then choose Block potassium conductance. The delayed outward potassium component disappears while the transient sodium component remains. These interventions reproduce the logic of current separation: hold the voltage command constant while selectively removing one modeled conductance.
The blocked-channel drawings are faded because the maximum conductance has been removed. This is a model intervention, not a claim that blocking a conductance leaves the rest of a biological axon unchanged.
Reverse the sodium gradient
Choose Reverse the sodium gradient (model experiment). The sodium reversal potential is moved to −40 mV while the command step goes to −20 mV. Sodium conductance can open normally, but now
\[
V-E_{\mathrm{Na}}>0,
\]
so the sodium current is outward rather than inward.
This is one of the most important demonstrations in the module. An open channel does not guarantee an inward current, and a large conductance does not guarantee a large current. Current depends on both conductance and electrochemical driving force.
Depolarized holding voltage
Choose Depolarized holding voltage (−50 mV). Before the test step even begins, the membrane has been held at a voltage that lowers \(h\), reducing sodium availability. Compare the gate state at the start of the command with the ordinary −65 mV holding condition.
This experiment anticipates refractoriness. The response to a test voltage depends on the state in which the membrane begins, and that state depends on its recent voltage history.
Voltage clamp turns voltage from a dependent variable into an experimental input. It reveals that sodium activation is fast, sodium inactivation is slower, and potassium activation is delayed. Those kinetic differences are the ingredients from which the action potential will be generated when voltage is released.
3. Action potential: interacting feedback processes
Select Action potential. The module now uses current clamp: applied current is controlled, and voltage is free to change. The full feedback loop is active:
\[
V\longrightarrow (m,h,n)
\longrightarrow (g_{\mathrm{Na}},g_{\mathrm{K}})
\longrightarrow (I_{\mathrm{Na}},I_{\mathrm{K}})
\longrightarrow V.
\]
A 1 ms current pulse begins at 5 ms. The default suprathreshold pulse evokes one action potential.
Just below threshold
Choose Just below threshold. A \(6\ \mu\mathrm{A/cm^2}\) pulse depolarizes the membrane but does not trigger a regenerative spike. The trajectory makes a graded excursion and returns to rest.
Watch the current-balance panel. Sodium current may increase, but not enough to create a self-amplifying rise in voltage. Outward and leak contributions regain control, and the perturbation decays.
Suprathreshold pulse
Choose Suprathreshold pulse. A \(10\ \mu\mathrm{A/cm^2}\) pulse evokes one spike. Slow playback to 0.5× or 0.25×, select Currents, and use Step during the upstroke.
The sequence is continuous, but several changes in dominance can be identified.
Initial membrane charging
The applied current first charges the membrane. Voltage rises modestly, shifting the \(m\) target upward. Sodium activation begins to increase.
Regenerative sodium feedback
As \(m\) rises, sodium conductance increases. Because \(V\) remains far below \(E_{\mathrm{Na}}\), sodium current is inward and depolarizing. That depolarization raises \(m\) still further, which increases sodium conductance again. This is fast positive feedback.
The response becomes regenerative when inward sodium contribution grows faster than the outward currents can oppose it. The stimulus does not draw the spike directly; it places the membrane in a state from which the membrane’s own dynamics generate the upstroke.
The spike peak
At the voltage maximum,
\[
\frac{dV}{dt}=0.
\]
The current-balance panel shows the depolarizing and hyperpolarizing contributions momentarily balancing. Sodium conductance has not necessarily vanished, and potassium conductance has not suddenly appeared. Both currents have been changing throughout the upstroke.
The sodium driving force also shrinks as \(V\) approaches \(E_{\mathrm{Na}}\). Thus, sodium current can weaken even while substantial sodium conductance remains.
Repolarization
Sodium availability \(h\) has been declining, while potassium activation \(n\) has been rising. The outward potassium contribution becomes dominant and voltage falls. Repolarization is therefore produced by overlapping sodium inactivation, potassium activation, and changing driving forces—not by one instantaneous channel switch.
Afterhyperpolarization and recovery
Potassium activation does not immediately return to its resting value when voltage crosses the resting level. Continued outward potassium current drives the membrane below rest. During recovery, \(n\) falls and \(h\) rises toward their resting targets. Excitability is restored gradually.
Stronger suprathreshold pulse
Choose Stronger suprathreshold pulse. Doubling the pulse to \(20\ \mu\mathrm{A/cm^2}\) generally shortens spike latency much more than it changes spike height. Once regenerative dynamics begin, the action-potential waveform is governed mainly by the conductances and reversal potentials, not by a voltage trace proportional to stimulus strength.
This does not mean that every property is invariant. A sufficiently different input or state can alter timing, trajectory, and recovery. The pedagogical point is that a suprathreshold pulse triggers an internally generated excursion rather than directly specifying the spike amplitude.
Block sodium conductance
Choose Block sodium conductance. The same current pulse now produces only a graded depolarization. Without voltage-dependent inward sodium feedback, there is no regenerative upstroke.
This is the clearest test of the positive-feedback mechanism.
Block potassium conductance
Choose Block potassium conductance. A holding current of \(-4.40\ \mu\mathrm{A/cm^2}\) keeps the membrane at \(-65\) mV, so the trial begins from the same voltage as the standard spike. The 1 ms pulse then evokes a regenerative event with a latency of about 1.05 ms and a peak near \(+49\) mV, higher than the normal \(+39\) mV because no outward potassium current opposes the upstroke.
What follows is the important part. The membrane does not repolarize. Sodium inactivation and leak pull it down from the peak, but it settles near \(-3.6\) mV and stays there for the rest of the trial. That is not a slow return to rest; it is the resting potential of a membrane that has lost its dominant resting conductance.
The resulting waveform is an intentionally abnormal model perturbation. It shows what potassium contributes to both the spike and the resting state. It should not be read as an ordinary action potential with one optional component omitted.
Block potassium without compensation
Choose Block potassium (no holding current). Nothing else changes; the bias current is simply removed.
The membrane leaves \(-65\) mV immediately and fires an unprompted action potential at about 2.8 ms — roughly 3.2 ms before the current pulse. The measurement box reports a negative latency, and the validation badge reports the failure explicitly rather than certifying the run.
This is worth doing deliberately, because it makes a point the compensated preset hides: \(-65\) mV is not a property of the membrane that survives an intervention. It is the voltage at which the particular set of conductances present happens to balance. Remove one of them and the equilibrium moves. The comparison between these two presets is the cleanest demonstration in the module that resting potential is a computed balance, not a setting.
Reduce the sodium reversal potential
Choose Reduce sodium reversal potential. The module sets \(E_{\mathrm{Na}}\) to 0 mV. Sodium activation can still occur, but the inward driving force is greatly reduced as voltage rises. The normal overshoot is prevented, and under this preset a regenerative spike does not occur.
This experiment separates the gating mechanism from the ionic gradient. Opening sodium conductance is not sufficient; the current must also have the appropriate direction and magnitude.
The named phases describe the voltage trace and the changing dominance of overlapping currents. Fast sodium activation, sodium inactivation, potassium activation, leak, capacitance, and driving force all operate continuously. The spike is the trajectory of their coupled interaction.
4. Refractory period: excitability has a history
Select Refractory period. Every experiment begins with a \(10\ \mu\mathrm{A/cm^2}\), 1 ms pulse at 5 ms. A second 1 ms pulse is delivered after a variable onset-to-onset interval.
The first spike leaves the membrane in a state different from rest:
- \(h\) is reduced, so less sodium conductance is available;
- \(n\) remains elevated, so outward potassium conductance is stronger;
- and \(V\) may already have returned near rest even though the gates have not.
The complete membrane state is therefore \((V,m,h,n)\), not \(V\) alone.
Absolute refractory condition
Choose Absolute refractory: early strong second pulse. The second pulse begins only 8 ms after the first and is doubled to \(20\ \mu\mathrm{A/cm^2}\). It still fails to evoke a second spike.
Pause at the second pulse and inspect \(h\) and \(n\). Sodium availability remains too low and potassium activation remains too high for a regenerative sodium excursion. The failure is not caused by an externally imposed “threshold” parameter. It emerges from the gate state left by the first spike.
Relative refractory condition: the original pulse fails
Choose Relative refractory: same pulse fails. The onset interval is now 12 ms, but the second pulse has the original \(10\ \mu\mathrm{A/cm^2}\) amplitude. It fails. The membrane has recovered more than in the absolute condition, but not enough for the original stimulus.
Relative refractory condition: a stronger pulse succeeds
Choose Relative refractory: stronger pulse succeeds. The interval remains 12 ms, but the second pulse is increased to \(20\ \mu\mathrm{A/cm^2}\). A second, altered spike is now evoked.
The comparison between the two 12 ms presets defines relative refractoriness operationally. The membrane is excitable, but it requires a stronger perturbation than it did at rest.
Recovered condition
Choose Recovered: same pulse succeeds. The onset interval is 20 ms, and the second pulse returns to \(10\ \mu\mathrm{A/cm^2}\). A second full action potential occurs because \(h\) and \(n\) have recovered sufficiently.
A useful way to inspect the experiment
Select Recovery variables, restart, and watch \(h\) and \(n\) between the two event lines. Then drag the timeline exactly to the second pulse and read the state boxes and the measurements labeled h at pulse 2 and n at pulse 2. Step through all four presets with ] and read the accumulated table in the measurement log:
| 8 ms |
0.368 |
0.497 |
no, even at 20 µA/cm² |
| 12 ms |
0.544 |
0.355 |
only at 20 µA/cm² |
| 20 ms |
0.605 |
0.313 |
yes, at 10 µA/cm² |
The two 12 ms rows are the useful pair: the gate state at the second pulse is identical, so the difference in outcome is attributable to stimulus amplitude alone.
The relevant comparison is not simply the voltage at which the second pulse begins. Two membranes can have similar voltages but different hidden gate states and therefore different responses to the same input.
A horizontal threshold line can be a rough descriptive landmark on a voltage trace, but it is not the mechanism that decides whether a spike occurs. Excitability depends on the membrane’s location in the multidimensional state space, including sodium availability and potassium activation inherited from recent activity.
5. Point attractor to limit cycle
Select Point to limit cycle. This mode connects the conductance-based membrane to the language of dynamical systems. The same four-dimensional Hodgkin–Huxley equations can display several qualitatively different trajectories depending on the initial state and applied current.
The state plot shows \(V\) versus \(n\). Color encodes \(h\), and the size of the moving point encodes \(m\). Faint reference paths show the four unperturbed experiments together so that their geometry can be compared.
Return to the resting fixed point
Choose Return to the resting fixed point. The simulation begins 5 mV above rest with no applied current. The small displacement decays and the trajectory returns to the resting state.
The resting state is a stable fixed point: once reached, the state no longer changes, and nearby states are drawn back toward it.
Subthreshold excursion
Choose Subthreshold excursion. A \(6\ \mu\mathrm{A/cm^2}\) pulse pushes the state away from rest, but the trajectory remains a small excursion and returns to the same fixed point.
This is more than a voltage statement. Every gate moves during the excursion, but the joint state remains within the basin of the resting attractor and does not enter the regenerative spike trajectory.
Single excitable spike
Choose Single excitable spike. A \(10\ \mu\mathrm{A/cm^2}\) pulse generates a large excursion through the projected state space and then returns to the same resting point.
A single action potential is therefore not a limit cycle. It is a large excitable transient. The trajectory does not continue around the loop after the pulse; it closes only in the loose visual sense that it returns to rest.
Sustained current and repetitive firing
Choose Sustained current: repetitive firing. A constant \(10\ \mu\mathrm{A/cm^2}\) current begins at 5 ms and remains present. After an initial transient, the membrane fires repeatedly with a nearly stable period. In the projected state plot, successive spikes traverse the same orbit.
The long-term behavior is now an attracting periodic orbit—a limit cycle. The state never becomes constant, but the pattern of change is stable. The measurement boxes report spike count, mean period, and the coefficient of variation of the period.
The difference from a single spike is fundamental:
| No input after a small displacement |
Return to rest |
Stable fixed point |
| Brief subthreshold pulse |
Small excursion, then rest |
Transient to fixed point |
| Brief suprathreshold pulse |
Large spike excursion, then rest |
Excitable transient to fixed point |
| Sustained current |
Repeated orbit |
Stable limit cycle |
Perturb the state
The Brief current nudge button is available in this mode. It adds a \(0.45\) ms, \(18\ \mu\mathrm{A/cm^2}\) current pulse just after the present model time. Pause or scrub to a chosen moment, press the button, and then resume playback. Clear nudges removes all manually added perturbations. If a nudge is placed late in a trial, the trial is automatically lengthened so the recovery is actually recorded rather than cut off at the end.
Under the fixed-point conditions, the nudge displaces the membrane and the trajectory returns toward rest. During repetitive firing, a nudge can shift the system to another location on the cycle or transiently distort the trajectory, but the recurrent conductance dynamics restore the repeating orbit.
The exact timing of a nudge matters because the membrane’s state changes around the orbit. The same perturbation delivered during the upstroke, repolarization, or recovery does not begin from the same \(V,m,h,n\) combination.
An attractor is a property of the joint equations, not an object stored in one channel or one variable. Resting and repetitive firing use the same membrane components. Their different long-term behaviors arise from the organization of the complete state under different input conditions.
A guided laboratory sequence
The following sequence can be used as an individual exercise, a classroom demonstration, or the basis of a worksheet.
Part A: Separate current from voltage
- Open Passive membrane → Baseline membrane.
- Before pressing Play, predict whether voltage will jump when current begins.
- Play the simulation and identify the period during which applied current and leak current are both present.
- Compare Larger capacitance with Larger leak conductance. For each, predict the change in response speed and amplitude before viewing it.
- Use Stronger current pulse to determine whether a passive membrane has an all-or-none response.
The key conclusion is that current changes charge, and voltage evolves according to capacitance and the net conductive current.
Part B: Separate conductance from current
- Open Voltage clamp → Large depolarizing step.
- View the gating variables and identify which gate changes first.
- View conductances, then ionic currents. Note that the same gate dynamics are being viewed at successive explanatory levels.
- Block sodium, then potassium, while preserving the same voltage command.
- Use Reverse the sodium gradient to demonstrate that an open sodium conductance can carry outward current.
The key conclusion is
\[
\text{current}=\text{conductance}\times\text{driving force}.
\]
Part C: Locate regenerative feedback
- Compare Just below threshold and Suprathreshold pulse.
- Slow playback and pause during the early depolarization.
- Step forward and backward across the upstroke while watching \(m\), sodium conductance, sodium current, and \(\mathrm{d}V/\mathrm{d}t\). The rate of change appears both in the state readout and under the Voltage and dV/dt trace view.
- Identify the point at which sodium-driven depolarization begins to amplify itself.
- At the spike peak, verify that the net current is approximately zero even though ionic conductances remain active.
The key conclusion is that a spike begins when voltage-dependent inward current creates positive feedback strong enough to outrun the opposing currents.
Part D: Identify the terminating processes
- In the standard spike, compare \(h\) and \(n\) during the upstroke, peak, and falling phase.
- Block potassium conductance. The previous run is pinned automatically, so the loss of repolarization can be read directly against the normal spike.
- Compare the compensated and uncompensated potassium blocks, and state what the holding current is doing.
- Reduce \(E_{\mathrm{Na}}\) and observe the consequence of reducing sodium driving force.
The key conclusion is that spike termination reflects both changing gates and changing driving forces. Sodium inactivation and delayed potassium activation are not events that begin only after the voltage peak.
Part E: Demonstrate state-dependent excitability
- Open the four refractory presets in order.
- Record spike count, pulse interval, \(h\) at pulse 2, and \(n\) at pulse 2. The measurement log collects these automatically as you step through the presets; Copy as CSV exports the table.
- Compare the two 12 ms conditions, which differ only in second-pulse amplitude.
- Explain why the same \(10\ \mu\mathrm{A/cm^2}\) pulse fails at 12 ms but succeeds at 20 ms.
The key conclusion is that recent activity changes the hidden state from which the membrane responds.
Part F: Distinguish transients from attractors
- View the resting fixed point, subthreshold excursion, and single spike.
- Explain why all three eventually have the same long-term state.
- View sustained repetitive firing and compare its projected trajectory with the single-spike path.
- Apply a nudge during rest and during repetitive firing.
- Describe what remains stable in each case: a state for the point attractor and an orbit for the limit cycle.
Questions to guide interpretation
- Why does applied current begin before passive membrane voltage reaches its maximum?
- Why does increasing capacitance slow the response without changing the eventual voltage expected from a sufficiently long constant current?
- Why does increasing leak conductance make the response both faster and smaller?
- Under voltage clamp, why can the ionic currents change while voltage remains constant?
- What is the difference between a gate’s present value and its \(x_\infty(V)\) target?
- Why does \(m\) produce a rapid effect while \(h\) and \(n\) contribute slower changes?
- How can sodium conductance be high while sodium current is small?
- How can sodium current become outward in the reversed-gradient experiment?
- What evidence distinguishes a suprathreshold action potential from a larger passive response?
- Why does a stronger suprathreshold pulse change spike latency more than spike height in this model?
- Why does the voltage peak occur when net current is zero rather than when sodium conductance is zero?
- Why is the ordinary division into depolarization and repolarization an incomplete mechanistic explanation?
- Why can two membranes at nearly the same voltage respond differently to the same pulse?
- What distinguishes absolute from relative refractoriness in the paired-pulse experiments?
- Why is a single spike a transient rather than a limit cycle?
- In repetitive firing, what is stable if membrane voltage never becomes constant?
- Why must the \(V\)-versus-\(n\) graph be called a projection rather than the full state space?
- What does recovery after a nudge demonstrate that an unperturbed trajectory does not?
- Why does removing \(\bar g_{\mathrm{K}}\) change the resting potential, and why does removing \(\bar g_{\mathrm{Na}}\) change it so much less?
- What is the holding current compensating for, and what would you conclude from the uncompensated preset if no one told you the baseline had drifted?
- Why is a spike better identified by the membrane’s own regenerative current than by a fixed voltage line?
Scientific scope and limits
This is the classical squid giant-axon model, not a complete model of every neuron. Mammalian cortical, thalamic, cerebellar, sensory, and autonomic neurons contain additional voltage-gated, calcium-dependent, modulatory, and compartment-specific conductances. The value of the Hodgkin–Huxley model is that it exposes the general logic of conductance-based excitation with unusual clarity.
The simulation uses one spatially uniform, space-clamped compartment. It demonstrates membrane excitation but not action-potential propagation along an axon. Propagation requires axial current between spatial compartments, so that local depolarization charges neighboring membrane and regenerates the spike at successive positions.
The model is deterministic. It does not simulate stochastic opening and closing of individual channels. That approximation is appropriate for the population-level conductances represented here, but channel noise can matter in small membrane patches and small neuronal processes.
The module deliberately avoids a permanent threshold line. Threshold can be a useful descriptive measurement, but the model shows why it is not a fixed switch. Whether a perturbation triggers a spike depends on the full state and its recent history.
The conductance-block, altered reversal-potential, and reversed-gradient presets are controlled model experiments. They isolate causal contributions but need not correspond to a simple, harmless biological manipulation.
Removing a conductance also moves the resting potential, because the resting potential is whatever voltage makes the remaining currents sum to zero. The current-clamp block presets therefore hold the membrane at \(-65\) mV with a small bias current so that the stimulus, rather than the drift, causes what you see. That bias is reported rather than hidden, and one preset omits it deliberately so the effect can be observed directly.
Complete gate-rate functions
With voltage in millivolts and time in milliseconds, the active modes use
\[
\alpha_m(V)
=
\frac{0.1(V+40)}{1-\exp[-(V+40)/10]},
\]
\[
\beta_m(V)=4\exp[-(V+65)/18],
\]
\[
\alpha_h(V)=0.07\exp[-(V+65)/20],
\]
\[
\beta_h(V)
=
\frac{1}{1+\exp[-(V+35)/10]},
\]
\[
\alpha_n(V)
=
\frac{0.01(V+55)}{1-\exp[-(V+55)/10]},
\]
and
\[
\beta_n(V)=0.125\exp[-(V+65)/80].
\]
For each gate,
\[
x_\infty(V)=\frac{\alpha_x(V)}{\alpha_x(V)+\beta_x(V)}
\]
and
\[
\tau_x(V)=\frac{1}{\alpha_x(V)+\beta_x(V)}.
\]
The apparent singularities in \(\alpha_m\) at \(V=-40\) mV and \(\alpha_n\) at \(V=-55\) mV are removable; the numerical implementation evaluates their limiting values.
Primary sources
The model and its experimental basis were developed in the classic squid giant-axon studies:
- A. L. Hodgkin and A. F. Huxley, “A quantitative description of membrane current and its application to conduction and excitation in nerve,” The Journal of Physiology 117 (1952): 500–544. Open the article
- A. L. Hodgkin, A. F. Huxley, and B. Katz, “Measurement of current–voltage relations in the membrane of the giant axon of Loligo,” The Journal of Physiology 116 (1952): 424–448. Open the article
- R. FitzHugh, “Impulses and physiological states in theoretical models of nerve membrane,” Biophysical Journal 1 (1961): 445–466. Open the article
The central lesson
The Hodgkin–Huxley model replaces the statement “the neuron reaches threshold and fires” with a mechanism. A perturbation changes voltage. Voltage moves the gating targets. Gates change conductances. Conductances and driving forces produce ionic currents. The net current changes voltage again.
Fast sodium activation supplies regenerative positive feedback. Sodium inactivation and delayed potassium activation supply slower negative feedback. Their unequal time courses create excitability, the spike waveform, refractoriness, recovery, and—under sustained drive—repetitive firing. The action potential is not located in one channel and is not passed from one stage to the next. It is a property of the evolving state of the complete membrane system.
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